34 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
distinct powers of Y, not exceeding the (s-1)". Let the surd T be 
a chief (see Def. 4) subordinate of Y, but not a subordinate of any 
other surd in f (py); its index being = and, by changing T, wherever 
it occurs in f (p) in any of its powers, into zT, z being an r“ root of 
unity, distinct from unity, let f (p), A, Y, A., &c., be transformed 
into F (py), B, U, B,, &c.; so that 
F@=B4B,U £B.U + &e. 
Then, if F (p)= (p), the terms, 
AU Nee iN Ye 5 SOCK, teres Renta oat AN) 
taken in same order, are equal to the ae 
BBO 8, Us hoe. e 1g SO) 
each to each; A being equal to B. 
For, since F (p) = f (p), we have 
(AEB) AGW it AL Wr BoB, US Se. =10ci:, te 
Hence (Cor. Prop. I) one or other of the following equations must 
subsist : 
AY =D(A—B), 
NS! WD) Wal We aioe scale Med 
Ae DEL My, 
where D ig an expression involving only such surds as occur in the 
° 
r) 
expressions A, B, A,, B,, &c., or are subordinates of Y or of U; al 
being a term in the series, Y , Y , &c., distinct from y ; and wi 
representing some term in the series, uit Ue. &e. But, since T is 
not a subordinate of any surd in f (p) except Y, the coefficients B, 
B,, &e., involve no surds different from those which enter into the 
coeflicients A, A,, &c.; and therefore involve only surds which are 
found in f (p). Also, since T is not subordinate to any of the sub- 
ordinates of Y, it follows that the subordinates of U are the same 
with those of Y. Hence D involves only such surds as occur in 
f (p). Therefore (Cor. 1, Def. 9,) the first and second of equations 
(4) ares inadmissible ; and the third must subsist. Adopting then the 
equation, A, Y =D 1Bbe 1 , we say that no other term in (1) than 
A,Y can be equal to the product of B,, re by an expression such as 
